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4answers
167 views

How do we know logic works? [duplicate]

Every time I read about a theory in mathematics, it usually starts with axiomatizing the most fundamental concepts that are going to be treated. Recently, I have started finding this troubling. In ...
2
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2answers
81 views

Are theorems like subroutines for math? [on hold]

I've been developing more appetite for math just lately, as I study electromagnetics to deepen my understanding of electric circuits and devices. I'm finding that doing derivations as exercises helps ...
3
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3answers
79 views

Infinite sums vs infinite unions

Why is it that For every set $S$, there exists a set $\bigcup S$. is something we take for granted (even though $S$ could be infinite), while For every sequence $a_1,a_2,\dots$ of numbers, ...
2
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3answers
77 views

Basic atoms in mathematics [closed]

Given the concepts '1', 'set' and 'sum' (and maybe 'point' for geometry), can you build the whole mathematics upon then? If not, what other basic atoms would you need?
10
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9answers
1k views

Is math an exact science? [closed]

The day before yesterday I talked with a friend of mine about math. He is also a PhD student like me, and in his opinion math cannot be consider an exact science, because the same statement could be ...
1
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0answers
38 views

Is it possible to be a frequentist and a subjectivist at the same time?

I'm trying to understand the differences between (1) Bayesian vs frequentist; and (2) subjectivist vs objectivist. So far my understanding (correct me if I'm wrong) is that: (1) Bayesian vs ...
3
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1answer
39 views

Differences between constructivism and formalism

What are the main differences between the formalism and constructivism in mathematics? Is there some theorem or axiom valid in formalism which isn't valid in constructivism and vice versa? Is the ...
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2answers
71 views

In “10 grams of salt”, is the unit “grams”?

The gram is a unit of mass, so "10 grams" has "grams" as the unit. "10 pounds" uses a different unit. So what is the "salt" in "10 grams of salt", if not a unit? In other words, what is the ...
1
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0answers
59 views

Any “exotic” foundations of mathematics? [closed]

There is a myriad of axiomatizations of set theory (a branch of mathematics obviously not at all identical with notorious ZFC) and other formal systems working with classes, categories and such. All ...
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3answers
55 views

Why does $p$ (is true) strictly agree with $p$ while $p$ (is false) strictly disagrees?

Let's make the truth table: $$\begin{array}{|c|c|c|} \hline p&(p) \text{ is true}&(p) \text{ is false}\\ \hline T&T&F\\ F&F&T\\\hline \end{array}$$ "$p$ is true" strictly ...
2
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2answers
56 views

Logical explanation why exponentiation operation is not commutative and associative

Considering Peano axioms we'll define addition, multiplication and exponentiation operations. We can then prove that addition and multiplication operations are commutative and associative. The proof ...
3
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1answer
49 views

If a theory $A$ can prove $B$ consistent relative to $A$, and $A$ is consistent, does $B$ have to be consistent?

Let's say we have two sets of axioms $A$ and $B$ such that $\mathsf{ZF} \subseteq A \subseteq B$, and from $A$ we can prove that if $A$ is consistent, then $B$ is consistent as well (that is, $A \...
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2answers
68 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
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0answers
26 views

Multiplying an object by time instead of dividing. What happens when?

Let "a" be apples, and "s" be seconds. a= 1 apple. a/s= 1 apple for every second (an accumulation of apples). a/s/s= 1 apple for every second, for every second (an acceleration of the accumulation ...
18
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2answers
1k views

What's going on with this 5-line proof of Fermat's Last Theorem? [duplicate]

I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author ...
1
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0answers
55 views

How to become fluent at reading math formulas? [closed]

As part of my studies, oftentimes I need to read research publications which contain mathematical formulas. Whenever I have to do that, I feel discouraged. Somehow I can not comprehend the ...
0
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1answer
50 views

The positive introspection axiom

I am studying modal logic with the textbook 'Reasoning about Knowledge' Fagin et al. 1995 The positive introspection axiom is taken as something that can be proved with the possible worlds model of ...
1
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2answers
50 views

Necessary truth of mathematical proposition.

Take from Possible world- an introduction to logic and its philosophy. p-21 Following quote provide us with necessary definition of what "logically necessary" or as far as i think "necessary truth" ...
0
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1answer
81 views

Can we take definability and existence as primitive notions of a theory?

One of my friend tries to develop an alternative viewpoint of Set Theory. For this he has taken the terms binary relation, set, existence and definability as primitive notions of his Set Theory. After ...
5
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6answers
360 views

Do different notations imply different properties of a number?

I had an argument with a friend of mine and I'd be glad if someone could clarify things a little bit. So, let's say we have an integer, eight or seventeen, for example, doesn't matter. It has all the ...
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1answer
72 views

A philosophical question on probability theory [closed]

This question is philosophical in nature. The example is taken from theology, but one may invent more examples, including these more scientific than mine. Nevertheless it is a valid mathematical issue....
2
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1answer
70 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
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2answers
129 views

When are quantities outside of the real numbers considered equal, and when do they exist?

I know of the complex number $i$ and it's existence as the result of invalid square rooting (the square root of negative one does not exist inside the real numbers), but other than complex numbers, ...
53
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10answers
5k views

Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
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0answers
25 views

Tensors as mathematical objects

Continuing my journey to understand Tensors, Maxwell's equations. Here is my current understanding. Is it correct? Tensors are mathematical objects, i.e., an entity in mathematical reality or a ...
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4answers
699 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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1answer
96 views

Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
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6answers
1k views

Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
2
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1answer
62 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
2
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0answers
69 views

Is Leibnizian calculus embeddable in first order logic?

We just published an article making what we feel is a plausible case in favor of an affirmative answer in Foundations of Science, see preprint here. The basic argument is that while such a requirement ...
2
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1answer
51 views

Have humans ever used the Log Scale convention in the past rather than the Linear one?

There are many examples where our senses are based off of log scales such as volume of a noise, ability to guess (i.e.) plus or minus a power of 10 with Fermi, and even when we measure pain on 1 to 10 ...
3
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3answers
146 views

On “why” questions in mathematics

In response to the question How would one be able to prove mathematically that $1+1 = 2$?, Asaf Karagila explains: In a more general setting, one needs to remember that $0,1,2,3,…$ are just ...
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2answers
83 views

On logic vs information theory

If the statements All crows are black and All non black things are non crows are equal, then why is the former so much easier to communicate by giving examples? What implications does this ...
27
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9answers
2k views

Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
3
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0answers
71 views

Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? (I'm ...
1
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1answer
46 views

Different Mathematics

Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he ...
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1answer
26 views

A question about the real line and the Dirichlet function.

Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points). Here's my reasoning: ...
4
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4answers
105 views

Should we or should we not take $1$ as a prime number? [duplicate]

I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime ...
2
votes
3answers
42 views

Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
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0answers
38 views

mathematical terms with fractions and variables - usage in daily life?

what usage do algebraic fractions (monomial or polynomial) have in our life? Are there specific professions that deal with them now and then? Where exactly in technology do they have their very own ...
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0answers
9 views

$Con(T) + T\vdash \neg\neg A$ implies $Con(T+A)$ for any intuitionistic theory T

It's easy to notice that for any intuitionistic theory T: $Con(T) + T\vdash \neg\neg A$ implies $Con(T+A)$ $Con(T) + T\vdash \neg\forall x\neg A$ implies $Con(T+\exists x A)$ where $Con(T)$ means, ...
3
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2answers
63 views

Scalar multiplication as a special form of matrix multiplication

Question What do we gain or lose, conceptually, if we consider scalar multiplication as a special form of matrix multiplication? Background The question bothers me since I have been reading about ...
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2answers
56 views

Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
1
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1answer
105 views

Quasi mathematical objects [closed]

I was looking on this post http://www.songho.ca/math/euler/euler.html and I came to the comment that says "i is not a number at all. It is an ill-formed concept. There is a vast difference between a ...
2
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1answer
39 views

Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of Identity'...
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2answers
59 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
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1answer
75 views

Developments from Charles Peirce's logic diagrams?

These last weeks I have been revisiting Charles Sanders Peirce's logical or thought diagrams (what he called, alpha, beta and gamma diagramms) and I found many of them highly interesting. Some ...
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3answers
168 views

What exactly are the numbers we use everyday?

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its ...
2
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2answers
95 views

What is the simplest mathematical object? [closed]

What is the simplest mathematical object? I am talking about mathematics in the most abstract way possible, and not as some concrete axiomatic theory (e.g. foundational ones, like ZFC). After a lot ...
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0answers
98 views

How can I learn Math intuitively? [closed]

I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher ...